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Normal frames for derivations and linear connections and the equivalence principle. (English) Zbl 1023.53012

The work can be considered as a review, continuation and application of the following five papers: T. L. Agafonova and B. G. Kanaev, On a two-dimensional distribution in a four-dimensional projective space, Geometry of generalized spaces, 3-12, Penz. Gos. Ped. Inst., Penza (1992); B. Z. Iliev, J. Phys. A, Math. Gen. 29, 6895-6901 (1996; Zbl 0899.53064); B. Z. Iliev, Linear transports along paths in vector bundles. V. Properties of curvature and torsion. Commun. Joint. Inst. Nucl. Res. Dubna, E5-97-1; B. Z. Iliev, Transports along maps in fibre bundles, Commun. Joint. Int. Nucl. Res. Dubna, E5-97-2; A. V. Paramonov, Mat. Strukt. Model. 5, 23-30 (2000; Zbl 0956.53008). The study of existence, uniqueness and holonomity of normal frames on a manifold \(M\) with connection \(\Gamma\) reduces to the matrix differential equation \(XA= -\Gamma A\), for one vector field \(X\), for vector fields tangent to a submanifold or for any vector field on \(M\), and is linked with the properties of curvature.
Remark: There are some intersections with M. Rahula, New problems in differential geometry Singapore: World Scientific (1993; Zbl 0795.53002), see pp. 26-32.

MSC:

53B05 Linear and affine connections
53B50 Applications of local differential geometry to the sciences
57R25 Vector fields, frame fields in differential topology
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